and in the subsequent, to add the calibrated noise \(\xi _t\sim \mathcal {N}(0,\sigma ^2)\)  with \(\sigma = \frac{C}{\mu }\) . Thereby, the DP-SGD updates at the \(t\) -th step is given by [1]}:
\(\text{DP-SGD:} \quad \begin{split}\theta _t = \theta _{t-1} - \eta \frac{1}{\left|X_{t}\right|} \left(\sum _{x\in X_{ t}}\widetilde{g}_x+ \xi _{t}\right), \quad t\in [T].\end{split}\) 
